Errors, Variances and Correlations

We are so accustomed to thinking in terms of raw scores, with their apparent pin-point precision, that we find the concept of measurement error at first abhorrent and later perplexing. We understand, finally, that measurement error always exists, but we don't know how to handle it when reporting our results.

Measurement error always affects our findings. When reporting our results we need to decide whether the result is supposed to describe the local details of this data collection, or to infer some more general "reality". Local results, such as an individual's performance on a test and the test's reliability, must contain their measurement error. General inferences, however, such as the math level of third graders, are intended to be estimated without bias from measurement error. To ascertain what, if anything, of global import we have discovered, we need to deduct the effect of measurement error from our final estimates.

Consider person n with measure bn of standard error sn. The more observations we make, the more we know about this person, and the smaller sn will be. What about the dispersion of a sample of N people? When measurement error is uncorrelated with what is measured, the variance of the sample of observed measures contains the sample variance and also the measures' error variance. But it is only the sample variance that estimates the population sampled.

The estimation for this corrected sample variance is:

s^2 = sum( bn^2 - b.^2 - sn^2 ) / (N-1)

where b. = sum(bn)/N

Correlation coefficients are also affected. Measurement error reduces the maximum observable correlation. Adjusting observed correlation coefficients for measurement error isolates how much sample variance is left unexplained.

Consider the typical regression analysis, in which dependent measures are mistaken for points without error. Let R^2 be the observed multiple squared correlation (proportion of explained variance), E be the observed RMS residual from the regression, and S be the observed standard deviation of the measures. Then E contains both unexplained variance and measurement error, and R^2 has been reduced accordingly. R^2 is defined as

R^2 = (S^2 -E^2) / S^2

If r^2 is the squared multiple correlation corrected for measurement error in the dependent variable and e is the RMS measurement error, then

r^2 = (S^2 - E^2) / (S^2 - e^2)

= R^2 / [ 1 - (e/E)^2 (1 - R^2)]

e<=E; r>=R; when e=0, then r=R; when e=E, then r=1.

It is this larger correlation r, and not R, which tells us how successfully our independent variables have explained the variance of the dependent measure.

Errors, Variances and Correlations, B Wright … Rasch Measurement Transactions, 1991, 5:2 p. 147

Rasch Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene	Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver	Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone	Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind	Rasch Measurement: Applications, Khine	Winsteps Tutorials - free Facets Tutorials - free	Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre	Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan
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Rasch Measurement Transactions & Rasch Measurement research papers - free	An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse	Rasch Measurement Theory Analysis in R, Wind, Hua	Applying the Rasch Model in Social Sciences Using R, Lamprianou	El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias	Diseño de Mejores Pruebas - free, Spanish Best Test Design	A Course in Rasch Measurement Theory, Andrich, Marais	Rasch Models in Health, Christensen, Kreiner, Mesba	Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen

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